To find the area of triangle ABC with vertices A(8, 2), B(6, 4), and C(8, 10), we can use the formula for the area of a triangle given by the vertices' coordinates:
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Substituting the coordinates into the formula:
- \(A(8, 2) \rightarrow (x_1, y_1) = (8, 2)\)
- \(B(6, 4) \rightarrow (x_2, y_2) = (6, 4)\)
- \(C(8, 10) \rightarrow (x_3, y_3) = (8, 10)\)
Now, we calculate:
\[ \text{Area} = \frac{1}{2} \left| 8(4 - 10) + 6(10 - 2) + 8(2 - 4) \right| \]
Calculating each term in the absolute value:
- \(8(4 - 10) = 8 \times (-6) = -48\)
- \(6(10 - 2) = 6 \times 8 = 48\)
- \(8(2 - 4) = 8 \times (-2) = -16\)
Now, substituting back into the equation:
\[ \text{Area} = \frac{1}{2} \left| -48 + 48 - 16 \right| = \frac{1}{2} \left| -16 \right| = \frac{1}{2} \times 16 = 8 \]
The area of the triangle is \(8\) square units.
Given that Luella multiplies \(12\) by \(2\) and then by another number to find the area, we can see that:
\[ 12 \times 2 \times \text{(another number)} = 8 \]
Calculating what that "another number" would be:
\[ 12 \times 2 = 24 \]
To find the "another number," set it up as follows:
\[ 24 \times \text{(another number)} = 8 \]
Now, solving for "another number":
\[ \text{(another number)} = \frac{8}{24} = \frac{1}{3} \]
Thus, the number Luella can use to complete the calculation is:
\[ \boxed{\frac{1}{3}} \]